Mark detection method, exposure method, device manufacturing method, mark detection apparatus, exposure apparatus, and device

ABSTRACT

The object of the present invention is to provide a mark detection method, an exposure method, a device manufacturing method, a mark detection apparatus, an exposure apparatus, and a device manufactured by the use of the exposure apparatus, which are capable of reducing a position measurement error in a short time even in the case where a sampling interval must be set to about 0.2 times or more of a lower limit of a minimum periodic component. In the present invention, a mark formed on an object is irradiated with a detection beam, an image of the mark is picked up through an image-forming system, the image of the mark formed on an image pickup device is converted into an electrical image signal, a signal related to the image signal is output at predetermined sampling intervals, and a signal related to the image signal is interpolated in a cycle equal to or less than the predetermined sampling interval.

TECHNICAL FIELD

The present invention relates to a mark detection method, an exposuremethod, a device manufacturing method, a mark detection apparatus, andan exposure apparatus. More particularly, the present invention relatesto a mark detection method for detecting a mark for position measurementthat is formed on an object such as a semiconductor substrate or aliquid crystal display device, an exposure method for transferring apredetermined pattern on a substrate aligned by the use of the markdetection method, a device manufacturing method using the exposuremethod, a mark detection apparatus for detecting a mark for positionmeasurement which is formed on an object such as a semiconductorsubstrate or a liquid crystal display device, an exposure apparatus fortransferring a predetermined pattern onto a substrate aligned by themark detection apparatus, and a device manufactured by the exposureapparatus.

BACKGROUND ART

In manufacturing a semiconductor device and a liquid crystal displaydevice, a variety of planar techniques are utilized. In the planartechniques, a finely patterned image formed on a photomask and a reticle(hereinafter, referred to as reticle) by the use of an exposureapparatus is projected and exposed on a substrate such as asemiconductor wafer or glass plate on which a photosensitive agent suchas photoresist is coated (hereinafter, referred to as wafer).

The reticle pattern is projected and exposed by the use of, for example,an exposure apparatus of a step-and-repeat system in such a manner thata position of the reticle and a position of the wafer are adjusted(aligned) with high accuracy and the reticle pattern is superposed on apattern already formed on the wafer.

Particularly in recent years, high densification has been required forsemiconductor circuits. Accordingly, also in the alignment of theexposure apparatus, as the pattern of the semiconductor circuit or thelike becomes finer, a demand for an alignment performed with higheraccuracy has increased, and various processes for alignment have beenmade.

In general, the alignment of the reticle is performed using exposurelight.

Among alignment systems for the reticle, there is a Visual ReticleAlignment (VRA) system or the like, in which an alignment mark drawn ona reticle is irradiated with exposure light, and image data of thealignment mark picked up by a CCD camera or the like is subjected toimage processing, and a mark position is measured.

The following are types of alignment sensors for wafers.

(1) Laser Step Alignment (LSA)

This sensor is a sensor for irradiating alignment marks arranged as aline of dots on a wafer with a laser beam in order to detect a markposition by the use of light diffracted or scattered by the mark.

(2) Field Image Alignment (FIA)

This sensor is a sensor for irradiating alignment marks arranged as aline of dots with light having a large wavelength bandwidth using ahalogen lamp or the like as a light source, and performing imageprocessing of the image data of an alignment mark imaged by a CCD cameraor the like in order to measure a mark position.

(3) Laser Interferometric Alignment (LIA)

This sensor is a sensor for irradiating alignment marks arranged in adiffraction grating pattern on a wafer from two directions using laserbeams having slightly different frequencies and causing the twogenerated diffraction lights to interfere with each other in order tomeasure a position of the alignment mark from the phase obtained throughthe interference.

In the alignment by these optical systems, first, an alignment mark onthe reticle is detected and processed to measure a position coordinatethereof. Next, an alignment mark on the wafer is detected and processedto measure a position coordinate thereof, thus position of a shot to besuperposed is determined. Based on these results, the wafer is moved bya wafer stage to perform an alignment so that a pattern image of thereticle can be superposed on the shot position, and the pattern image ofthe reticle is projected and exposed on the wafer.

In some of the above-described alignment systems, processing isperformed after a one-dimensional image or a two-dimensional image isobtained as an alignment signal.

For the case of a two-dimensional image, by adding mark portions in themeasured direction, it can also be treated as a one-dimensional image.

These signals are originally signals that are continuously distributedwith respect to the position, but for convenience of signal transmissionof an image pickup device, the signals will be extracted as signalssampled at a predetermined interval. For example, when an imageprocessing sensor, such as a CCD camera or a line sensor, is used as animage pickup device, since the pixel size is limited, the signals willbe sampled at an interval determined by the pixel size. Ideally, it isdesirable that signals output from the image pickup device be sampled bya sampling apparatus at an interval corresponding to the pixel size ofthe image pickup device.

Edge detection, a correlation method or the like is used for thesesampled signals in order to measure the mark position.

Incidentally, in general, the accuracy required for the alignment sensoris extremely high in comparison to the minimum resolving unit of theimage pickup device. For this reason, the position must be finallydetermined with an accuracy equal to or less than the sampled interval.

Heretofore, in edge detection and the correlation method, processing hasbeen performed for the sampled signals, and when the final positionresult is calculated, the interval between the sampled points is fittedby an appropriate function such as a linear or a quadratic function, andby solving the function, a resolving power less than the sampledinterval has been obtained. Typically, the finer the sampling interval,the more the accuracy is improved.

On the other hand, when the magnification of the optical system isincreased in order to reduce the sampled interval on an object, thevisual field is narrowed due to a limitation in the number of pixels ofthe CCD camera.

Considering the constitution of the apparatus, the visual field of thesensor must be ensured to some extent by conditions such as size of thealignment mark or the accuracy of the pre-alignment performed before thealignment measurement.

In addition, in order to prevent the conversion of a high-frequencycomponent of a signal into a low-frequency component by sampling(aliasing), a necessary condition is that the minimum resolving unit ofthe image in the image pickup device is 0.5 times or less of the minimumperiodic component.

The minimum periodic component of the signal is given by, for example,in the case of an image processing sensor using an optical microscope,the lower limit of the minimum periodic component of the image asP_(min) as follows:

P _(min)=λ(2×NA)

λ: wavelength of light

NA: NA of optical system

However, this value will also vary depending on the illuminationconditions. By using this value, the sampling interval P_(s) is givenas:

P _(s)<0.5×P _(min)

and, the above-described conditions can be satisfied.

However, when the sampling interval P, is increased, the error insampling when performing the edge measurement or correlation measurementbecomes significantly worse before the sampling interval P_(s) reaches0.5×P_(min), that is, from about 0.2×P_(min).

FIG. 14 is a diagram for explaining the process during the execution ofthe edge detection.

In a typical edge detection algorithm, first, the point of maximuminclination, slope point is found. Typically, since the samplinginterval is a fixed value, a point is obtained where a difference ΔVbetween adjacent sampled points in the V direction in the drawing is ata maximum. In the example of FIG. 14, the point denoted by the referencesymbol P₀ is the point of maximum inclination.

From this point, the closest relative maximum and minimum are found byhill-climbing and hill-descending. In the example shown in FIG. 14, withreference symbol P₀ as a center, points are found in the H₁ and H₂directions where the difference ΔV in the V direction in the drawingbecomes a minimum. These points are defined to be the maximum value andthe minimum value of the edge. In the example of FIG. 14, the sampledpoint P₁ becomes the maximum value of the edge, and the sampled point P₂becomes the minimum value of the edge.

After the maximum value and the minimum value of the edge are obtained,setting a slice level SL as, for example, an intermediate value of thesevalues, the edge positions E₁ and E₂ are set as the points where theedge crosses the slice level.

When the sampling interval is increased to some extent, the maximumvalue and the minimum value of the above-described edge will varyaccording to the relationship between the sampled position and thesignal edge position. For this reason, the slice level SL varies,resulting in a variation of the measurement result.

In addition, since the fitting is performed by a linear function, aquadratic function, or the like, when obtaining the edge positions E₁and E₂, an error occurs here also.

Also in the correlation method, depending on the positional relationshipbetween the sampled position and the signal, the mark signal causesdeformation so as to change the center of gravity thereof, resulting ina variation in the measurement result.

Moreover, also in the correlation method, since a resolving power whichis smaller than the sampled interval results from the fitting to aquadratic or the like, an interpolation error occurs here also.

Furthermore, heretofore, in order to improve the accuracy, a pluralityof marks have been typically used for the alignment marks, and theaccuracy has also been improved by averaging of the respective marksobtained by sampling in different phases.

Incidentally, when performing superposition by an exposure apparatus, ashift of the image according to the structure of the alignment mark andresulting from a comatic aberration becomes a problem. However, it hasrecently been found that the comatic aberration can be improved byincreasing NA.

In addition, since the measurement accuracy improves as the edge slopebecomes steeper for a signal which has a noise at the same level, it isalso necessary to increase NA of the optical system used in thealignment in order to improve the alignment accuracy.

However, when NA of the alignment optical system is increased, a problemoccurs in that the minimum periodic component P_(min) included in theimage, that is,

P _(min)=λ(2×NA)

decreases as NA increases.

Since it is difficult to narrow the visual field under the presentsituation, it has become difficult to satisfy the condition ofP_(s)<0.2×P_(min).

In addition, in using an XY-simultaneous mark for measuring X and Ysimultaneously in order to increase speed (refer to the gazette ofJapanese Unexamined Patent Application, First Publication No. Hei2-272305 for details), the number of alignment marks must be reduced,and thus the above-described averaging effect also decreases.

Hereinbelow, simulation results will be used to describe therelationship between the sampling interval and the position measurementerror.

FIGS. 15 and 16 are diagrams showing position measurement errors whensampling is performed for a step difference mark of a 6 μm line fordifferent sampling intervals. FIG. 15 is a diagram showing positionmeasurement errors when imaging is performed with an optical systemhaving a wavelength of 0.6 μm, an illumination sigma=1, and NA=0.6. FIG.16 is a diagram showing position measurement errors when imaging isperformed with an optical system having a wavelength of 0.6 μm, anillumination sigma=1 and NA=0.3. In FIGS. 15 and 16, the abscissa showssampling intervals, and the ordinate shows position measurement errors.

In this simulation, the minimum periodic component P_(min) included inthe above-described image is 1 μm when NA is 0.3 and 0.5 μm when NA is0.6.

As shown in FIGS. 15 and 16, when the sampling interval is changed, aresult is obtained wherein the position measurement errors arecyclically reduced. However, this sampling interval in which theposition measurement errors are reduced changes depending on the linewidth.

In alignment of the exposure apparatus and the like, the total requiredoverlay (total overlay accuracy) also varies depending on the line widthof the circuit pattern printed on the substrate. However, for this totaloverlay, typically an accuracy of ¼ or less of the minimum line width ofthe printed circuit pattern is required. An accuracy of about 50 nm istypically required for the total overlay. In order to satisfy thisrequirement, the measurement error allowable for the alignment sensor isabout 3 to 5 nm.

Now, the alignment error allowable for alignment is assumed to be 5 nm.

As is apparent from FIGS. 15 and 16, even if NA is 0.6 and 0.3, when thesampling interval P_(s) is 0.2×P_(min)<P_(s)<0.39×P_(min), and when thesampling interval P_(s) is P_(s)>0.41×P_(min), the point measurementerror exceeds the allowable error.

FIG. 17 is a diagram showing the simulation results representing therelationship between the sampling interval and the position measurementerror when a normalized mutual correlation is used.

As shown in FIG. 17, as the sampling interval P_(s) becomes longer, theposition measurement errors smoothly increase, and, with aboutP_(s)<0.2×P_(min) as a reference, the deterioration of the accuracycannot be permitted.

Next, the results obtained by performing the edge detection with anincrease in the number of alignment marks will be shown.

FIGS. 18 to 20 are diagrams showing the simulation results representingthe relationship between the sampling interval and the positionmeasurement error when the number of alignment marks is changed. FIG. 18is a diagram showing the result when the number of line spaces(hereinafter referred to as L&S) is three; FIG. 19 is a diagram showingthe result when L&S is six; and FIG. 20 is a diagram showing the resultwhen L&S is nine.

In general, as the number of alignment marks is increased, the positionmeasurement error decreases. However, from the results shown in FIGS. 18to 20, it is understood that there is a sampling interval where theposition measurement error does not decrease very much even if thenumber of alignment marks is increased.

Moreover, with reference to FIGS. 18 to 20, sampling intervalscyclically appear at which the position measurement errors are extremelyreduced. The magnification of the optical system may be set so that thesampling interval thereof can be matched to the above-described samplinginterval. However, this is not so desirable because the magnificationaccuracy during manufacturing needs to be made strict and different markintervals cannot be dealt with.

DISCLOSURE OF THE INVENTION

The present invention was made in consideration of the foregoingcircumstances in mind. The object of the present invention is toprovide, even if a sampling interval must be set to about 0.2 times ormore of the lower limit of the minimum periodic component, a markdetection method capable of reducing a position error of a markposition, an exposure method for transferring a predetermined patternonto a substrate aligned using the mark detection method, a devicemanufacturing method using the exposure method, a mark detectionapparatus, an exposure apparatus for transferring a predeterminedpattern onto a substrate aligned using the mark detection apparatus, anda device manufactured by the use of the exposure apparatus.

Another object of the present invention is to provide, even if thesampling interval is equal to 0.5 times or more of the lower limit ofthe minimum periodic component, that is, even if the sampling intervaldoes not satisfy a sampling theorem, a mark detection method capable ofpreventing a position measurement error caused by aliasing, an exposuremethod for transferring a predetermined pattern onto a substrate alignedusing the mark detection method, a device manufacturing method using theexposure method, a mark detection apparatus, an exposure apparatus fortransferring a predetermined pattern onto a substrate aligned using themark detection apparatus and a device manufactured using the exposureapparatus.

In order to accomplish the foregoing objects, a first mark detectionmethod of the present invention comprises the steps of: irradiating amark formed on an object with a detection beam; imaging an image of themark through an image-forming system; converting the image of the markwhich is formed on an image pickup device into an electrical imagesignal; outputting a signal related to the image signal in apredetermined sampling interval; and interpolating the signals relatedto the image signal in cycles equal to or less than the predeterminedsampling intervals.

According to this invention, in the case where the minimum resolutionunit of an image pickup device must be 0.2 times the minimum periodiccomponent of an image or more, there is an advantage that the positionmeasurement error of a mark can be significantly reduced byinterpolating in a cycle equal to or less than a predetermined samplinginterval for image signals of the mark which are sampled at thepredetermined intervals.

In addition, in the above-described mark detection method, the imagepickup device has a predetermined pixel size, the predetermined samplinginterval includes an interval of the predetermined pixel size, and theinterpolation is performed at an interval equal to or less than thepredetermined pixel size. In this case, the pixel size is preferably apredetermined multiple of the minimum periodic component of the imageformed on the image pickup device. This minimum periodic component isdefined by λ/2NA based on the wavelength λ of the detection beam and thenumerical aperture NA of the imaging system. Moreover, it is preferablethat the pixel size be 0.2 times the minimum periodic component or moreand, further, 0.5 times the minimum periodic component or less. In thiscase, it is best if the pixel size is anywhere between 0.39 times ormore to 0.41 times or less of the minimum periodic component. Moreover,it is preferable to perform a smoothing operation to remove a componentequal to or less than a predetermined cycle from the image signal outputat the sampling intervals.

Furthermore, it is preferable that the smoothing operation remove aperiodic component equal to or less than 1/(1/P_(s)−1/P_(min)), which isrepresented by the predetermined pixel size P_(s) and the minimumperiodic component P_(min).

By performing this processing, aliasing noise can be removed in the casewhere the minimum resolving unit P_(s) of the image pickup device mustbe 0.5 times or more of the minimum periodic component P_(min) of theimage; thus there is an advantage that error accuracy can be improvedeven in conditions where the image of an object cannot be completelyrestored.

As a result, since the position measurement in a coarse samplinginterval can be performed on, for example, the image of the alignmentmark, the expansion of NA or the expansion of the visual field ispossible even by the use of a conventional image pickup device.

Moreover, since an interpolation operation accompanied with smoothing isperformed in this invention, there is an advantage in that processingtime can be shortened as compared with the case where interpolation andlow pass filtering are performed separately.

Furthermore, the predetermined pixel size P_(s) is preferably more than0.5 times the minimum periodic component P_(min).

Specifically, the smoothing operation comprises: a step of setting asmoothing point where smoothing is performed for the image signal; astep of selecting, from the image signal, an image signal sampled in apredetermined range that includes the smoothing point; a step ofsampling a function while removing a periodic component smaller than1/(1/P_(s)−1/P_(min)) according to the distance between the position ofthe smoothing point and a position of the selected image signal in acycle identical to the sampling interval of the image signal; and a stepof adding the product of the selected image signal and the sampledfunction, the product being obtained for each of the image signalsincluded in the predetermined range. Alternatively, the smoothingoperation comprises: a step of setting an interpolation point where theinterpolation for the image signal is to be performed; a step ofobtaining a most proximate position of the image signal, which is mostproximate to the position of the interpolation point; a step ofselecting, from the image signal, an image signal sampled in apredetermined range including the most proximate position; and a step ofadding the product of the selected image signal and the functionremoving a periodic component smaller than 1/(1/P_(s)−1/P_(min))according to the distance from the position of the selected imagesignal, the product being obtained for each of the image signalsincluded in the predetermined range.

Moreover, in this invention, the image signal is output as a samplepoint in the predetermined sampling interval, and an interpolation isperformed on an arbitrary point in a cycle equal to or less than thepredetermined sampling interval by an interpolation method using aconversion including the linear combination of a plurality of the samplepoints located in the vicinity of the arbitrary point. Thisinterpolation method includes a weighting operation using the pluralityof the sample points.

Furthermore, in this invention, the position of the object is measuredon the basis of the interpolated image signal. Herein, the predeterminedsampling interval is determined on the basis of the amount of positionmeasurement error in the measurement. The object predetermined herein isa substrate onto which a circuit pattern is transferred, and the amountof position measurement error in the predetermined sampling interval isthe value where a total overlay becomes ¼ or less of the minimum linewidth of the circuit pattern transferred onto the substrate.

Still further, in this invention, the interpolation is performed on theimage signal itself and on a correlation function obtained on the basisof the image signal.

In order to accomplish the foregoing objects, a second mark detectionmethod of the present invention comprises the steps of: imaging a markformed on an object; converting an image of the mark, which is formed onan image pickup device, into an electrical image signal; outputting asignal related to the image signal in a predetermined sampling intervalas a sample point; and performing an interpolation on an arbitrary pointin a cycle equal to or less than the predetermined sampling interval byan interpolation method using a conversion including the linearcombination of a plurality of the sample points.

Herein, in the interpolation method, the arbitrary point is subjected tointerpolation using the plurality of the sample points located in thevicinity of the arbitrary point. This interpolation method includes astep of performing a weighting operation using the plurality of thesample points, and an interpolation filter for determining a weightingcoefficient used in the weighting operation. Herein, when thepredetermined sampling interval is T, the interpolation filter includesan interpolation function s(dx) given by as:${s({dx})} = \frac{\sin ( {2\quad \pi \quad {{dx}/2}T} )}{2\quad \pi \quad {{dx}/2}T}$

The interpolation filter is represented as: S(dx)=s(dx)·W(dx). S is aproduct of the interpolation function s(dx) and a window function W(dx)which is capable of converging an end portion of the interpolationfunction s(dx) to zero. Herein, when a length of the window is R, thewindow function W(dx) is represented as:${W({dx})} = \frac{1 + {\cos ( {2\quad \pi \quad {{dx}/R}} )}}{2}$

Moreover, the second mark detection method of the present inventionstandardizes the interpolation filter so that the sum total of theweighting coefficients used when a first arbitrary point is subjected tointerpolation and the sum total of the weighting coefficients used whena second arbitrary point different from the first arbitrary point issubjected to interpolation can be predetermined values. Herein, thestandardization converts the respective coefficients of theinterpolation filter by dividing the respective coefficients by the sumtotal of the respective coefficients.

Furthermore, the second mark detection method of the present inventionperforms a smoothing process for removing a component having a cycleequal to or less than a predetermined cycle from the image signal outputas a sample point in the predetermined sampling interval. Herein, theimage pickup device has a pixel size P_(s) a predetermined multiple ofthe minimum periodic component P_(min) of the image formed on the imagepickup device. The smoothing process includes a step of removing aperiodic component equal to or less than 1/(1/P_(s)−1/P_(min)), on thebasis of the pixel size P_(s) and the minimum periodic componentP_(min). The pixel size P_(s) is larger than 0.5 times the minimumperiodic component P_(min).

Furthermore, the interpolation is performed on the image signal itself.

The exposure method of the present invention is characterized in thatthe object is a substrate onto which a predetermined pattern istransferred, and the predetermined pattern is transferred onto thesubstrate which is aligned on the basis of the mark detected by the useof the mark detection method.

Moreover, a device manufacturing method of the present invention is formanufacturing a device using the exposure method of transferring thepredetermined pattern onto the substrate.

A first mark detection apparatus of the present invention includes anirradiation system which irradiates a mark formed on an object with adetection beam, an image-forming system which forms an image of the markon the image-forming surface, and an image pickup device disposed abovethe image-forming surface, a sampling device which converts the image ofthe mark into an electrical image signal in order to output a signalrelated to the image signal in a predetermined sampling interval, and aninterpolation device which interpolates the signal related to the imagesignal in a cycle equal to or less than the predetermined samplinginterval.

Herein, the first mark detection apparatus of the present invention ischaracterized in that the image pickup device has a pixel size apredetermined multiple of the minimum periodic component of an imageformed on the image-forming surface, the predetermined sampling intervalincludes a cycle of the pixel size, and the interpolation deviceperforms an interpolation in a cycle equal to or less than the pixelsize. The minimum periodic component is defined by λ/2NA based on thewavelength λ of the detection beam and the numerical aperture NA of theimage-forming system. The pixel size is preferably 0.2 to 0.5 times theminimum periodic component. In addition, the first mark detectionapparatus of the present invention further comprises a smoothing devicewhich removes a component equal to or less than a predetermined cyclefrom a signal output at the sampling intervals by the sampling device.Herein, the pixel size P_(s) is larger than 0.5 times the minimumperiodic component P_(min), and the smoothing device removes periodiccomponents equal to or less than 1/(1/P_(s)−1/P_(min)), on the basis ofthe pixel size P_(s) and the minimum periodic component P_(min).

Moreover, the first mark detection apparatus of the present invention ischaracterized in that the sampling device outputs a signal related tothe image signal as a sample point in the predetermined samplinginterval, and the interpolation device interpolates an arbitrary pointin a cycle equal to or less than the predetermined sampling interval byan interpolation method using a conversion including the linearcombination of a plurality of the sample points located in the vicinityof the arbitrary point.

Furthermore, the first mark detection apparatus further comprises ameasurement device which measures a position of the object on the basisof the interpolated signal, characterized in that the predeterminedsampling interval is determined on the basis of the amount of positionmeasurement error of the measurement by the measurement device.

Still further, the first mark detection apparatus is characterized inthat the sampling device outputs the image signal itself in thepredetermined sampling interval, and the interpolation device performsan interpolation on the image signal itself.

A second mark detection method of the present invention comprises asampling device which images a mark formed on an object, converting animage of the mark into an electrical image signal and outputting asignal related to the image signal in a predetermined sampling intervalas a sampling point; and an interpolating device which interpolates anarbitrary point in a cycle equal to or less than the predeterminedsampling interval by an interpolation method using a conversionincluding the linear combination of a plurality of the sample points.Herein, the interpolation device interpolates the arbitrary point usingthe plurality of the sample points located in the vicinity of thearbitrary point, and the interpolation device performs a weightingoperation using the plurality of the sample points, and includes aninterpolation filer which determines a weighting coefficient used in theweighting operation. When the predetermined sampling interval is T, thisinterpolation filter includes an interpolation function s(dx)represented as:${s({dx})} = \frac{\sin ( {2\quad \pi \quad {{dx}/2}T} )}{2\quad \pi \quad {{dx}/2}T}$

Moreover, the first mark detection apparatus of the present inventionfurther comprises a standardizing device which standardizes theinterpolation filter so that the sum total of the weighting coefficientsused when interpolation is performed on a first arbitrary point and thesum total of weighting coefficients used when interpolation is performedon a second arbitrary point different from the first arbitrary point canbe predetermined values. This standardizing device converts therespective coefficients of the interpolation filter by dividing therespective coefficients by the sum total of the respective coefficients.

Furthermore, the first mark detection apparatus of the present inventionfurther comprises a smoothing device which removes a component equal toor less than a predetermined cycle from the signal output as a samplepoint in the predetermined sampling interval by the sampling device.Herein, the sampling device includes an image pickup device having apixel size P_(s) of a predetermined multiple of the minimum periodiccomponent P_(min) of an image formed on a predetermined image-formingsurface, and the smoothing device removes a periodic component equal toor less than 1/(1/P_(s)−1/P_(min)), on the basis of the pixel size P_(s)and the minimum periodic component P_(min). This pixel size P_(s) ispreferably larger than 0.5 times the minimum periodic component P_(min).

In addition, the second mark detection apparatus of the presentinvention is characterized in that the sampling device outputs the imagesignal itself in the predetermined sampling interval, and theinterpolation device performs an interpolation on the image signalitself.

The exposure apparatus of the present invention is characterized in thatthe object is a substrate onto which a predetermined pattern istransferred, and the predetermined pattern is transferred onto thesubstrate which is aligned on the basis of the mark detected by the useof the mark detection apparatus.

Furthermore, a device of the present invention is manufactured throughthe step of transferring the predetermined pattern onto the substrate bythe exposure apparatus.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a diagram schematically showing a constitution of an exposureapparatus of one embodiment of the present invention.

FIG. 2 is a diagram for explaining a state in which interpolation isperformed on a one-dimensional image.

FIG. 3 is a diagram for explaining a state in which interpolation isperformed on a two-dimensional image signal.

FIG. 4 is a one-dimensional projection of a step difference mark imageformed on an image pickup device by an optical system for conditionswhere NA is 0.6, the wavelength is 0.6 μm, and the illumination sigma is1.0.

FIG. 5 is a signal obtained by sampling the image shown in FIG. 4 with asampling interval of 0.24 μm.

FIG. 6 is a signal obtained by sampling the image shown in FIG. 4 with asampling interval of 0.24 μm.

FIG. 7 is a diagram showing the dislocation amount of an edge positionwhen an image position for an image pickup device is moved.

FIG. 8 is a diagram showing the simulation results obtained byperforming a three-point interpolation on the respective sampled pointsfor the signal shown in FIG. 5.

FIG. 9 is a diagram showing the simulation results obtained byperforming a three-point interpolation on the respective sampled pointsfor the signal shown in FIG. 6.

FIG. 10 is a diagram showing the dislocation amount of an edge positionwhen an image position for an image pickup device subjected tointerpolation is moved.

FIG. 11 is a diagram showing the simulation results of positionmeasurement error when a sampling interval is changed when performingonly interpolation.

FIG. 12 is a diagram showing the simulation results of positionmeasurement error when the sampling interval is changed when performingonly low-pass filtering.

FIG. 13 is a diagram showing the simulation results of positionmeasurement error when the sampling interval is changed when performinginterpolation and low-pass filtering.

FIG. 14 is a diagram for explaining the process of edge detection.

FIG. 15 is a diagram showing position measurement error when a stepdifference mark of a 6 μm line is sampled for different samplingintervals.

FIG. 16 is another diagram showing the position measurement error when astep difference mark of a 6 μm line is sampled for different samplingintervals.

FIG. 17 is a diagram showing the simulation results representing therelationship between the sampling interval and position measurementerror when normalized mutual correlation is used.

FIG. 18 is a diagram showing the simulation results representing therelationship between the sampling interval and position measurementerror when the number of alignment marks is three.

FIG. 19 is a diagram showing the simulation results representing therelationship between the sampling interval and position measurementerror when the number of alignment marks is six.

FIG. 20 is a diagram showing the simulation results representing therelationship between the sampling interval and position measurementerror when the number of alignment marks is nine.

FIG. 21 is a flowchart for the production of a device (a semiconductorchip such as an IC or LSI, a liquid crystal panel, a CCD, a thin-filmmagnetic head, a micro-machine or the like) in the embodiment of thepresent invention.

FIG. 22 is a flowchart showing the details of step S304 of FIG. 21 forthe case of a semiconductor device.

BEST MODE FOR CARRYING OUT THE INVENTION

Hereinbelow, a description will be made in detail for a mark detectionmethod, an exposure method, a device manufacturing method, a markdetection apparatus, and an exposure apparatus according to oneembodiment of the present invention with reference to the drawings.

Exposure Apparatus

First, a description according to one embodiment of the presentinvention will be made for an exposure apparatus, to which a markdetection method according to one embodiment of the present invention isapplied.

FIG. 1 is a diagram schematically showing a constitution of the exposureapparatus according to one embodiment of the present invention.

In FIG. 1, reference numeral 1 denotes a light source such as asuper-high-pressure mercury lamp or an excimer laser. Reference numeral4 denotes a reflection mirror for reflecting illumination light emittedfrom light source 1. Reference numeral 5 denotes a wavelength selectionfilter which only transmits light having a wavelength necessary forexposure therethrough. Reference numeral 6 denotes a fly-eye integrator6 which adjusts the illumination light transmitted through wavelengthselection filter 5 into a luminous flux having a uniform intensitydistribution.

Reference numeral 7 denotes a reticle blind having an aperture S, whichadjusts the illumination range of the illumination light for a reticle10, to be described later, by changing the size of aperture S.

Wavelength selection filter 5, fly-eye integrator 6 and reticle blind 7are disposed sequentially on the same optical axis C1.

Reference numeral 8 denotes a reflection mirror for bending the opticalaxis C1, and reference numeral 9 denotes a lens system for irradiating areticle with the illumination light reflected by reflection mirror 8.

Reticle 10 is disposed on an optical axis C2 where lens system 9 isdisposed. On this reticle 10, a shot pattern is transferred onto a wafer12 and an alignment mark for position measurement, which are to bedescribed later.

Reference numeral 11 denotes a projection optical system disposed onoptical axis C2, which converges the illumination light transmittedthrough reticle 10.

Wafer 12 is a semiconductor substrate made of silicon or the like, andresist (not shown) is coated on a surface thereof.

Reference numeral 13 denotes a stage for holding wafer 12 by vacuumsuction. In addition, the stage 13 has a known structure in which a pairof blocks movable in directions perpendicular to each other aresuperposed onto each other. Reference numeral 21 denotes a drive devicesuch as a motor, which moves stage 13 in a stage-moving coordinatesystem formed in directions perpendicular to each other. Therefore,drive device 21 moves stage 13, and thus the shot position on wafer 12is superposed on an exposure visual field of projection optical system1.

In addition, a moving mirror 14 is fixed on a predetermined position ofstage 13. Reference numeral 20 denotes a laser interferometer whichdetects the position of stage 13 in the stage-moving coordinate systemby irradiating moving mirror 14 disposed on stage 13 with a laser beam15.

The above-described drive device 21 and laser interferometer 20 arecontrolled by a stage control system 36.

In addition, a reference mark member 33 having a height equal to thesurface of wafer 12 is fixed on a predetermined position of stage 13. Ona surface of reference mark member 33, a mark as a reference foralignment is formed. By measuring this mark, a reference position of thealignment sensor can be determined, and a positional relation betweenstage 13 and reticle 10 can be measured.

Stage control system 36 controls drive device 21 based on a controlsignal output from a main control system 37 to control movement of stage13.

In addition, detection results of laser interferometer 20 are suppliedfrom stage control system 36 to main control system 37, and main controlsystem 37 outputs a control signal to stage control system 36 based onthe information of the results.

The exposure apparatus in this embodiment comprises: a reticle alignmentsensor 31 for detecting an alignment mark formed on reticle 10; and awafer alignment sensor 32 for detecting a reference mark on referencemark member 33 or an alignment mark formed on wafer 12.

Reticle alignment sensor 31 detects the alignment mark of reticle 10 bythe use of, for example, light (detection beam) having a wavelengthequal to that of the illumination light (exposure light) from a lightsource (irradiation system, not shown). The relative position of adetection origin (for example, the center of an index mark) of reticlealignment sensor 31 and an optical axis AX of the projection opticalsystem are obtained beforehand. And the relative position of the centerof a circuit pattern region drawn on the reticle and the alignment markis also obtained beforehand. Accordingly, the center of the circuitpattern of the reticle and optical axis AX of the projection opticalsystem can be aligned by detecting the alignment mark and by obtainingan amount of a shift thereof from the detection origin.

Wafer alignment sensor 32 is a so-called off-axis alignment sensor whichis separately provided outside projection optical system 11. On an imageprojection surface, the optical axis of wafer alignment sensor 32 andthe optical axis of projection optical system 11 are parallel to eachother. Wafer alignment sensor 32 detects the reference mark provided onreference mark member 33 or the alignment mark formed on wafer 12, andthus measures the relative positional relation between these detectedmarks and the index mark formed in alignment sensor 32. Note that thereference mark or the alignment mark is irradiated with the detectionbeam from a light source (irradiation system) which is not shown.

Reticle alignment sensor 31 and Wafer alignment sensor 32 compriseimage-forming systems which form images of the alignment mark and thereference mark and the like on an image pickup device.

In addition, although all types of the above-described sensors areapplicable as wafer alignment sensor 32, an example of this embodimentwill be described using an FIA alignment sensor as wafer alignmentsensor 32.

Reference numeral 35 denotes an alignment control system to whichreticle alignment sensor 31 and the wafer alignment sensor 32 areconnected. Alignment control system 35 processes alignment signalsoutput from alignment sensors 31 and 32, and outputs the same to theabove-described main control system 37.

In the foregoing constitution, the illumination light emitted from lightsource 1, such as a super-high-pressure mercury lamp or an excimerlaser, is reflected by reflection mirror 4 and incident on wavelengthselection filter 5. Wavelength selection filter 5 only transmits lighthaving a wavelength necessary for exposure therethrough, and theillumination light transmitted through wavelength selection filter 5 isadjusted into a luminous flux having a uniform intensity distribution byfly-eye integrator 6 and then reaches reticle blind 7.

Reticle blind 7 changes the size of aperture S to adjust theillumination range on reticle 10 by the illumination light. Theillumination light transmitted through aperture S of reticle blind 7 isreflected by reflection mirror 8 and incident on lens system 9. Theimage of aperture S of reticle blind 7 is formed on reticle 10 by thislens system 9, and the desired range of reticle 10 is illuminated.

The image of the shot pattern or the alignment mark, which exists in theillumination range of the reticle 10, is formed on wafer 12 coated withresist by projection optical system 11, and thus the pattern image ofreticle 10 is exposed on a particular region of wafer 12.

Reticle alignment sensor 31 detects the position of the alignment markformed on reticle 10, and wafer alignment sensor 32 detects thereference mark position on reference mark member 33 fixed on stage 13,and thus the alignment signal is output. This alignment signal is outputto alignment control system 35, and reference positions of reticle 10and stage 13 are set.

Next, main control system 37 outputs a control signal to stage controlsystem 36 so that the alignment mark formed on wafer 12 can be detectedby the wafer alignment sensor 32. Stage control system 36 drives stage13 on the basis of this control signal, supplies a detection signaloutput from laser interferometer 20 to main control system 37, andcontrols the detection signal by sending feedback to it.

In such a manner, main control system 37 controls and moves stage 13,measures the position of the alignment mark formed on wafer 12, measuresthe positions of reticle 10 and wafer 12, and performs an alignment forthe same. At the point when the alignment is terminated, the shotpattern formed on reticle 10 is transferred to the resist coated onwafer 12.

A description will be made in detail of the procedure for determining analignment mark position from the alignment signals output from reticlealignment sensor 31 and wafer alignment sensor 32.

It is effective to perform the interpolation of the minimum divisionunit of the image pickup device if it is 0.2 to 0.5 times the minimumperiodic component of the image formed on the image pickup device by thealignment optical system.

The mark image formed on the image pickup device is sent as anelectrical signal to a sampling apparatus which is a sampling device,and sent as a digital image signal to a processing system. Thisprocessing system constitutes an interpolation device, a smoothingdevice, a standardizing device and a measurement device.

A line sensor, a CCD camera or the like can be used as the image pickupdevice. In the former case, the digital image signal is one-dimensional,and in the latter case, two-dimensional.

In addition, it is desirable that the sampling interval for the digitalimage signal output from the sampling apparatus typically be made tocorrespond with the minimum division unit of the image pickup device. Insuch a manner, the resolving power of the image pickup device can beefficiently utilized.

Mark Detection Method

Next, a description will be made in detail of a mark detection methodaccording to one embodiment of the present invention.

Interpolation of an Image Signal

First, a description will be made of a method for performing aninterpolation of a digital image signal obtained by the foregoing imagepickup device.

To simplify the description, the case of a one-dimensional image will bedescribed. Even a two-dimensional image can be treated as aone-dimensional image by removing one scanning line or by addingscanning lines in one direction.

FIG. 2 is a diagram for explaining a state in which interpolation isperformed on a one-dimensional image. In FIG. 2, for simplicity, adescription will be made for the case of an interpolation where an imageis treated as a one-dimensional image.

A case is considered in which interpolation is performed on aone-dimensional digital image signal f_(k)(k:0, 1, . . . , n) having asampling interval T in a cycle T′ satisfying T′<T. The signal afterbeing subjected to interpolation is defined to be f′_(k′)(k′:0, 1, . . ., n′). In order to perform the interpolation, it is satisfactory that aconvolution operation is performed using an interpolation filter S(k,k′) by the use of a point K, where a position X(k) on a digital imageindicated by k and a position X′(k′) indicated by k′ are coincident witheach other or proximate to each other (hereinafter, referred to as aproximate point), and several points in front of and behind this point.

A description will be made more specifically with reference to FIG. 2.In FIG. 2, f₃ to f₆ denote one-dimensional digital image signals, eachhaving a sampling interval T. The position coordinates of theseone-dimensional digital image signals f₃ to f₆ are X₃ to X₆,respectively.

Now, the case is considered in which the interpolation is performed onthese one-dimensional digital image signals in the cycle T′. In the casewhere the interpolation is performed between the one-dimensional digitalimage signals f₄ and f₅, the signal after the interpolation is definedto be f′₈ and the position coordinate thereof is defined to be X′₈.

A one-dimensional digital image signal proximate to the positioncoordinate of the signal after the interpolation is f₅. Thisone-dimensional digital image signal f₅ corresponds to the proximatepoint K, which has a value of K=5.

Therefore, the interpolation is performed by the use of several pointsin front of and behind the proximate point K, for example, using theone-dimensional digital image signals f₄ and f₅ and the followingEquation (1): $\begin{matrix}{{f^{\prime}k^{\prime}} = {\sum\limits_{({{K - m} < k < {K + m}})}{f_{k_{4}}{S( {k,k^{\prime}} )}}}} & (1)\end{matrix}$

In the foregoing Equation (1), the interpolation filter S(k, k′) is afunction of k and k′ in general; however, in the case of performing aposition measurement, the interpolation filter S(dx), which depends onlyon the relative position dx=|X(k)−X(k′)|, is typically used.

Moreover, when the sampling intervals T and T′ are in an integer ratio,the values that the relative position dx can take are limited. All ofthese numerical values are stored beforehand held in a storage apparatusand by reading these numerical values from the storage apparatus duringthe interpolation, the interpolation operation can be performed. Notethat the variable “m” used in Equation (1) is a variable determined inconsideration of the accuracy of the interpolation and the like.

Furthermore, in FIG. 2, in the case of obtaining f′₇, since the positioncoordinate of X′₇ and the position coordinate X₄ of the one-dimensionaldigital image signal f₄ are the same, f₄ corresponds to the proximatepoint K and the value thereof is K=4.

When a sampling function represented by Equation (2) is used as theforegoing interpolation filter S(dx), the original signal cantheoretically be restored. $\begin{matrix}{{S({dx})} = \frac{\sin ( {2\quad \pi \quad {{dx}/2}T} )}{2\quad \pi \quad d\quad {x/2}T}} & (2)\end{matrix}$

However, an errors occurs which is attributable to the area of the pixelof the image pickup device and the use of only data adjacent to thesignal in the interpolation.

The error caused by the latter case noticeably appears when theinterpolation filter S(dx) does not smoothly converge to zero at the endportions thereof.

For this reason, in actually, the product of the interpolation filterS(dx) having a limited length and the length itself, that is, a windowfunction W(dx) matching the distance fixed by the variable m consideredin performing the interpolation, is used as the interpolation filterS(dx). The following Equation (3) is an equation representing thisinterpolation filter S(dx). $\begin{matrix}{{S({dx})} = {\frac{\sin ( {2\quad \pi \quad {{dx}/2}T} )}{2\quad \pi \quad {{dx}/2}T} \times {W({dx})}}} & (3)\end{matrix}$

As an example of the foregoing window function, a panning window or thelike represented by the following Equation (4) may be satisfactorilyused. $\begin{matrix}{{{W({dx})} = \frac{1 + {\cos ( {2\quad \pi \quad {{dx}/R}} )}}{2}}{R\text{:}\quad {range}\quad {and}\quad {length}\quad {of}\quad {the}\quad {window}}} & (4)\end{matrix}$

In the above-described method, for the case where the positioncoordinate X(k′) is coincident with the position coordinate X(k), all ofthe points become zero except for the point where the relative positiondx=0. Accordingly, the data prior to the interpolation may be used as itis for the relative position, and thus the processing speed may beincreased.

In addition, although in the above description, the case was describedwherein an image signal obtained from the image pickup device is handledas a one-dimensional image, the processing can be performed for the casewherein an image signal is handled as a two-dimensional image.

Removal of Noise

Moreover, there are many cases where noise is included in the imagesignal which is a processing object. This noise is electrical noisegenerated on the periphery of the image signal or a defect, such as dustor the like, caused when the alignment mark is subjected to processing.

Particularly, in the case of performing asymptotic processing such asedge detection, high-frequency noise causes detection errors and thelike, resulting in large measurement error.

For this reason, in the case of performing a position measurement, ingeneral, low-pass filtering is commonly performed.

As described above, the interpolation is performed on the image signalobtained in this embodiment, and thereafter, in the case of filtering,the filtering is performed at the sampling intervals after theinterpolation. When the interpolation is performed, the samplinginterval after the interpolation becomes shorter than the samplinginterval for the image signal. Accordingly, in the case where thefiltering is performed after the interpolation, many operations arerequired and the processing time increases.

Furthermore, although the filtering can be performed before theinterpolation, in this case, the process of filtering becomes noise inthe interpolation operation, resulting in a deterioration of theaccuracy.

However, the interpolation operation as described above is a convolutionoperation which is basically the same as the filtering operation.Therefore, it is considered that the deterioration of the accuracy isprevented by simultaneously performing the interpolation operation, andthat the filtering operation and the processing speed can be increased.

Now, a predetermined low-pass filter is assumed to be R(x).

In the case of performing low-pass filtering after the interpolation isperformed using the interpolation filter S(x) represented in theabove-described Equation (2) or (3), first, the processing is performedon the image signal using the interpolation filter S(x), and then theprocessing is performed using the low-pass filter R(x).

Although not equivalently to the above process, when the followingEquation (5) is used, an almost similar process can be performed.

LPS(x)=∫S(x′)*R(x−x′)dx′  (5)

Therefore, the image signal obtained from the image pickup device isprocessed by the use of LPS(x) represented in Equation (5), and thus asignal subjected to interpolation and filtering can be obtained; andsince these processes are performed simultaneously, the processing timecan be shortened.

An operation for the case where a sinc function is used as theabove-described low-pass filter, R(x) corresponds to a change of thecycle of the sinc function of the interpolation filter S(x) representedby Equation (2).

Accordingly, in this case, LPS(x) in Equation (5) can be represented asin the following Equation (6) by the use of a variable H satisfying H<T.$\begin{matrix}{{{LPS}({dx})} = \frac{\sin ( {2\quad \pi \quad {{dx}/2}H} )}{2\quad \pi \quad {{dx}/2}H}} & (6)\end{matrix}$

Moreover, in the case of using the window function W(dx) represented inEquation (3), LPS(dx) is represented as in the following equation (7).$\begin{matrix}{{{LPS}({dx})} = {\frac{\sin ( {2\quad \pi \quad {{dx}/2}H} )}{2\quad \pi \quad {{dx}/2}H} \times {W({dx})}}} & (7)\end{matrix}$

By using Equation (6) or (7) in the processing of the image signalobtained from the image pickup device, the interpolation and low-passfiltering can be performed at a higher rate.

The foregoing window function is represented by the following Equation(8). $\begin{matrix}{{{W({dx})} = \frac{1 + {\cos ( {2\quad \pi \quad {{dx}/R}} )}}{2}}{R\text{:}\quad {range}\quad {and}\quad {length}\quad {of}\quad {the}\quad {window}}} & (8)\end{matrix}$

Interpolation of a Two-dimensional Image Signal

Next, a description will be made in detail for the case where the imagesignal obtained from the image pickup device is a two-dimensional image.

In the case of performing position measurement using a two-dimensionalimage, position coordinates in two directions may be obtained from thesame image data, or a position coordinate in only one direction may beobtained.

In this embodiment, for the case where the minimum division unit of theimage pickup device is about 0.2 times the minimum periodic component ofan image in at least one measurement direction, the interpolation isperformed so that the sampling interval of the digital image can besmaller than the minimum division unit of the image pickup device atleast in this direction.

A case is considered wherein the interpolation is performed on atwo-dimensional image signal f_(kx, ky) (sampling interval T_(x) in thex direction, sampling interval T_(y) in the y direction) and thetwo-dimensional image signal f_(kx, ky) is converted into a signalf_(kx′, ky′)(kx′:0, 1, . . . , nx′; ky′:0, 1, . . . , ny′) sampled in acycle T_(x)′ and T_(y)′ satisfying at least one of T_(x)>T_(x)′ andT_(y)>T_(y)′.

After interpolation, each value of the image signal f_(kx′, ky′) isobtained by the following Equation (9) using signals f_(kx, ky) of oneor more limited close points (P_(x)−mx<kx<P_(x)+nx,P_(y)−my<ky<P_(y)+ny) and an arbitrary interpolation filter S(kx, kx′,ky, ky′) depending on k and k′, with a point (P_(x), P_(y)) as a center,wherein a position P_(x)(kx), P_(y)(ky) on a two-dimensional imagecorresponding to kx, ky, and a position P_(x)′(kx′), P_(y)′(ky′)corresponding to kx′, ky′ are coincident or proximate to each other.Note that the foregoing variable m is a variable set in relation to theaccuracy. $\begin{matrix}{{f^{\prime}}_{k^{\prime}} = {\sum\limits_{{({{{Px} - {mx}} < {kx} < {{Px} + {nx}}})}{({{{Py} - {my}} < {ky} < {{Py} + {nx}}})}}{f_{k_{1}}{S( {{kx},{kx}^{\prime},{ky},{ky}^{\prime}} )}}}} & (9)\end{matrix}$

In order to facilitate understanding regarding the above-describedprocess, a specific example will be described with reference to FIG. 3.

FIG. 3 is a diagram for explaining a state in which the interpolation isperformed on a two-dimensional image signal. In FIG. 3, to simplify theexplanation, a case will be described wherein the sampling intervals ofthe two-dimensional image in two directions (x direction and y directionin FIG. 3) have the same T.

In FIG. 3, (P_(x)(2 x), P_(y)(3y)) to (P_(x)(6 x), P_(y)(6 y)) areassumed to be points sampled by the image pickup device.

Now, the case is considered where the interpolation is performed asampling interval T′ satisfying T′<T. Note that, in order to facilitateunderstanding, the case will be described wherein the interpolation isperformed at the sampling intervals T′ in any of the x direction and they direction.

As shown in FIG. 3, the sampled points after the interpolation aredefined to be (P_(x)′(3 x′), P_(y)′(5 y′)) to (P_(x)′(9 x′), P_(y)′(10y′)).

Now, the case is considered where the interpolation is performed for apoint Q of (P_(x)′(5 x′), P_(y)′(8 y′)) as shown in FIG. 3. First, aproximate point to the point Q to be interpolated (hereinafter, referredto as a proximate point) is obtained. In FIG. 3, (P_(x)(4 x), P_(y)(5y)) becomes the proximate point.

Next, at least one of the close points of this proximate point isselected. In the example shown in FIG. 3, points q1 to q8 are selected.After selecting these close points, the operation represented by theforegoing Equation (9) is performed.

The interpolation filter S(kx, kx′, ky, ky′) is a function depending onthe position in the two-dimensional image signal shown by k and theposition shown by k′, similar to the case where the interpolation isperformed on the one-dimensional image signal. In many cases, theinterpolation on the two-dimensional image signal is represented usingthe difference of these positions.

For the case where the distance of these positions in the x direction isdefined to be dx=|P_(x)(kx)−P_(x)′(kx′)| and the distance thereof in they direction is defined to be dy=|P_(y)(ky)−P_(y)′(ky′)|, the foregoinginterpolation filter S(kx, kx′, ky, ky′) is represented as theinterpolation filter S(dx, dy).

Moreover, the following is also similar to the case where theinterpolation is performed on the one-dimensional image signal.Specifically, if the sampling interval T and the sampling interval T′are in an integer ratio, when the operation of the foregoing Equation(9) is executed, the values of all the interpolation filters S(dx, dy)that the above-described variable can take are stored beforehand in thestorage apparatus and can be read while performing the interpolationoperation.

The interpolation filter S(dx, dy), for the case of performing theinterpolation on the two-dimensional image signal, becomes a product ofinterpolation filters s_(x) and s_(y) in their respective directions andis represented by the following Equation (10).

 S(dx,dy)=s_(x)(dx)×s _(y)(dy)  (10)

where${s_{x}({dx})} = \frac{\sin ( {2\pi \quad {{dx}/2}T_{x}} )}{{2\pi \quad {{dx}/2}T_{x}}\quad}$${s_{y}({dy})} = \frac{\sin ( {2{{\pi dy}/2}T_{y}} )}{{2\pi \quad {{dy}/2}T_{y}}\quad}$

The window function, for reducing noise due to the fact that theinterpolation filters s_(x) and s_(y) are limited filters, and thefiltering, for allowing the interpolation filters s_(x) and s_(y) tohave a low-pass effect or the like, are enabled in a manner similar tothe case of the one-dimensional image signal when each of all of theinterpolation filters s_(x)(dx) and s_(y)(dy) is subjected to the windowfunction and the low-pass filtering, and then a product thereof is used.

Moreover, similar to the case of performing noise removal for theone-dimensional image signal, with the interpolation filters s_(x)(dx)and s_(y)(dy) represented in Equation (10), H_(x) and H_(y) satisfying asampling interval of H_(x)<T_(x), H_(y)<T_(y) are used to convertEquation (10) into the following Equation (11), and thus low-passfiltering and interpolation can be performed simultaneously.$\begin{matrix}{{{s_{x}({dx})} = \frac{\sin ( {2\pi \quad {{dx}/2}H_{x}} )}{{2\pi \quad d\quad {x/2}H_{x}}\quad}}{{s_{y}({dy})} = \frac{\sin ( {2\pi \quad {{dy}/2}H_{y}} )}{{2\pi \quad {{dy}/2}H_{y}}\quad}}} & (11)\end{matrix}$

Furthermore, by obtaining a product of S(dx, dy) in Equation (10) andthe window function W(dx, dy), a deficiency wherein the value does notsmoothly converge to zero at the end portion of the interpolation filterduring noise removal may be resolved.

One example of the foregoing window function W(dx, dy) is given by thefollowing Equation (12).

W(dx,dy)=w_(x)(dx)×w _(y)(dx)  (12)

where${w_{x}({dx})} = \frac{1 + {\cos ( {2\pi \quad d\quad {x/R_{x}}} )}}{2}$${w_{y}({dy})} = \frac{1 + {\cos ( {2\pi \quad d\quad {y/R_{y}}} )}}{2}$Rx:  range  and  length  of  the  window  in  the  x  directionRy:  range  and  length  of  the  window  in  the  y  direction

Elimination of Aliasing (Smoothing)

In general, in order to obtain an original image from a signal obtainedby sampling after performing the sampling for a certain image, thesampling interval must be 0.5 times or less of the minimum cycleincluded in the image. Herein, for the case where the wavelength oflight used for the illumination is λ and the numerical aperture of theoptical system is NA, the minimum periodic component of the opticalsystem is λ/(2×NA).

However, in order that the alignment mark is first detected in theexposure apparatus, the magnification of the optical system is reducedto make the visual field broader. After the alignment mark is detected,the magnification of the optical system is increased and the alignmentis performed.

In such a case, the minimum division unit of the image pickup devicesometimes becomes 0.5 times or more of the minimum periodic component ofthe image formed on the image pickup device by the alignment opticalsystem. Hereinbelow, a description will be made for a mark detectionmethod capable of reducing the position measurement error even in such acase.

For the case where the foregoing conditions are not satisfied, forsampling performed by the image pickup device, aliasing occurs wherein ashort periodic component of the image is converted into a long periodiccomponent. For this reason, there is a problem that, in principle, asignal cannot be restored only by performing the interpolation.

In this embodiment, a pseudo-signal component due to aliasing isincluded in the signal and then removed.

When the minimum periodic component of the image pickup device isdefined as P_(s) and the minimum periodic component of the image formedon the image pickup device by the alignment optical system is defined asP_(min), the minimum periodic component of the pseudo-signal isrepresented as 1/(1/P_(s)−1/P_(min)).

Accordingly, filtering for removing the component having this cycle orless may be satisfactorily performed.

For this, it is satisfactory that an operation represented by thefollowing Equation (13) may be performed using a filter F(k, k′) whichdoes not include the periodic component approximately equal to or lessthan 1/(1/P_(s)−1/P_(min)). $\begin{matrix}{( f^{\prime} )_{k^{\prime}} = {\sum\limits_{({{K - m} < k < {K^{\prime} + m}})}{f_{k}{F( {k,k^{\prime}} )}}}} & (13)\end{matrix}$

Moreover, by obtaining the product of the right side of the aboveEquation (13) and the window function W(dx), a deficiency wherein thevalue does not smoothly converge to zero at the end portion of theinterpolation filter during noise removal may be resolved.

Furthermore, in the actual processing, values obtained by the operationrepresented by Equation (13) are obtained beforehand and stored in thestorage apparatus in tubular form, and during the interpolationoperation, the values may be read from the storage apparatus to performthe process. In such a manner, the operation processing time can beshortened, and further shortening of the processing time can beachieved.

Still further, a description will be made for the case where thelow-pass filtering of a two-dimensional signal f_(kx, ky) (kx:0, 1, . .. , nx; ky:0, 1, . . . , ny; nx, ny≧1; at least one of nx and ny is >1;x-directional sampling interval T_(x); y-directional sampling intervalT_(y)) is performed. Now, the signal after the low-pass filtering isgiven as f′_(kx)′_(ky)′.

First, as for each value f′_(kx)′_(ky)′, a point (P_(x), P_(y)) isobtained wherein a position P_(x)(kx), P_(y)(ky) on the digital imagewhich corresponds to kx, ky, and a position P_(x)′(kx′), P_(y)′(ky′)corresponding to kx′, ky′ are coincident or proximate to each other.

Next, with this point (P_(x), P_(y)) as a center, for signals f_(kx, ky)of one or more close points (P_(x)−mx<kx<P_(x)+nx,P_(y)−my<ky<P_(y)+ny), an operation shown by Equation (14) is performedon an arbitrary function S(kx, kx′, ky, ky′) depending on k and k′ andon a function F(x, y) which can be regarded as not including a periodiccomponent approximately smaller than 1/(1/P_(s)−1/P_(min)) in at leastone direction by the use of a numerical value row F_(i, j) obtained bysampling in the cycle T_(x), T_(y). Thus, low-pass filtering of thetwo-dimensional signal can be performed. $\begin{matrix}{{f^{\prime}}_{{kx}^{\prime},{ky}^{\prime}} = {\sum\limits_{{({{{Px} - {mx}} < {kx} < {{Px} + {nx}}})}{({{{Py} - {my}} < {ky} < {{Py} + {ny}}})}}{f_{{kx},{ky}}F_{{{kx} - {kx}^{\prime}},{{ky} - {ky}^{\prime}}}}}} & (14)\end{matrix}$

In addition, a description will be made for the case of performinginterpolation together with low-pass filtering.

The signal after converting the foregoing two-dimensional signalf_(kx, ky) is defined as f′_(kx)′_(ky)′(kx′:0, 1, . . . , nx′; ky′:0, 1,. . . , ny′). This signal after the conversion has a sampling intervalsatisfying at least one of T_(x)>T_(x)′ and T_(y)>T_(y)′.

In the case of performing interpolation together with low-passfiltering, first, for each value of f′_(kx)′_(ky)′, a proximate point(P_(x), P_(y)) is obtained wherein the position P_(x)(kx), P_(y)(ky) onthe digital image which corresponds to kx, ky, and the positionP_(x)′(kx′), P_(y)′(ky′) corresponding to kx′, ky′ are coincident orproximate to each other.

Next, with this point (P_(x), P_(y)) as a center, an operationrepresented in Equation (15) is performed using signal of one or morelimited close points f_(kx, ky) (P_(x)−mx<kx<P_(x)+nx,P_(y)−my<ky<P_(y)+ny) and an arbitrary function S(kx, kx′, ky, ky′)depending on k and k′, and thus low-pass filtering and interpolation ofthe two-dimensional signal can be performed. $\begin{matrix}{{f^{\prime}}_{k^{\prime}} = {\sum\limits_{{({{{Px} - {mx}} < {kx} < {{Px} + {nx}}})}{({{{Py} - {my}} < {ky} < {{Py} + {ny}}})}}{f_{k}{S( {{kx},{kx}^{\prime},{ky},{ky}^{\prime}} )}}}} & (15)\end{matrix}$

Herein, the function S(kx, kx′, ky, ky′) is a function s(dx, dy)depending on the variables dx=P_(x)(kx)−P_(x)′(kx′) anddy=P_(y)(ky)−P_(y)′(ky′). This function S(dx, dy) is a function capableof being regarded as approximately 1/(1/P_(s)−1/P_(min)) in at least anyone of the x direction and the y direction.

In addition, the function S(dx, dy) is represented by Equation (16)using H_(x) and H_(y) satisfying:

for the case of P_(s) {x}>0.5×P{x} _(max),

H _(x)=1/(1/P _(s)(x)−1/P{x}_(max));

for the case of P _(s) {y}>0.5×P{y} _(max),

H _(y)=1/(1/P _(s)(y)−1/P{y}_(max))

(where the symbol { } denotes the x or y direction.)

S(dx, dy)=s _(x)(dx)×s _(y)(dy)  (16)

where${s_{x}({dx})} = \frac{\sin ( {2\pi \quad {{dx}/2}H_{x}} )}{{2\pi \quad d\quad {x/2}H_{x}}\quad}$${s_{y}({dy})} = \frac{\sin ( {2\pi \quad {{dy}/2}H_{y}} )}{{2\pi \quad {{dy}/2}H_{y}}\quad}$

Moreover, as for the foregoing function S(dx, dy), one obtained byfurther multiplying S(dx, dy), represented by Equation (16), by thewindow function W(dx, dy) may also be used.

Furthermore, in the actual processing, a method may be also adopted suchthat values of the operation in Equation (16) are obtained beforehandand stored in the storage apparatus in tabular form, and during theinterpolation operation, values corresponding to the value of dx and dyare read from the storage apparatus and processed. With such a method,the operation processing time can be shortened and further shortening ofthe time can be achieved.

It is recommended that the filter F_(kx−kx)′, _(ky−ky)′ be made as alimited length filter F(dkx, dky) depending, respectively, on kx−kx′ andky−ky′. There are various types of filters, but obtained by multiplyinga sinc function by an appropriate window function W(dx) is used, forexample.

When the minimum periodic component of a false signal is represented asP_(c)=1/(1/P_(s)−1/P_(min)), the filter F(dx) is represented by Equation(17). $\begin{matrix}{{F({dx})} = \frac{\sin ( {2{\pi d}\quad {x/P_{c}}} )}{2\pi \quad {{dx}/P_{c}}}} & (17)\end{matrix}$

Other than the above, for a Savitzky-Golay filter, for example, as thefilter size thereof is made larger, the minimum periodic component isrelatively increased. Therefore, a similar effect can be obtained by theuse of a filter having an appropriate size.

In the above description, elimination of aliasing for a one-dimensionalimage signal was described. However, similar to the case ofinterpolation, such elimination can also be achieved for atwo-dimensional image signal using the product of the filter functionsobtained in their respective directions.

Moreover, low-pass filtering and interpolation can be performedsimultaneously. The method of performing noise removal simultaneouslywith interpolation was described previously. However, if the conditionis satisfied wherein the interpolation filter does not approximatelyinclude a periodic component equal to or less than1/(1/P_(s)−1/P_(min)), even for the case where the minimum division unitof the image pickup device is 0.5 times or more of the minimum periodiccomponent of the image, a measurement with good accuracy is possible.

The interpolation operation has been described as above. Hereinbelow, adescription will be made for the use of interpolation in the positionmeasurement.

Edge detection and a correlation method are well-known as mark delectionmethod.

In edge detection, an edge is predetermined by calculating adifferential image signal of an input image, a slice level is fixed forthe edge by a predetermined method, and the position where the edgecrosses the slice level is interpolated therein; thus a measurement isperformed with accuracy equal to or less than the sampling interval ofthe digital image data or less.

An example of the correlation method is a normalized mutual correlationmethod, in which a template image signal is prepared beforehand, anormalized mutual correlation between the template image signal and aninput image signal is calculated while shifting the relative position ofthe template image signal and the input image signal, and a correlationfunction depending on this relative position is computed; thus a peakposition of the correlation function is obtained.

Moreover, there is a self-correlation method used for the case wherethere is symmetry in the input image.

In the self-correlation method, a signal is divided into two parts at aposition considered to be a close point to a symmetrical point, theposition coordinate of one part of the divided signal is inverted, and acorrelation is calculated.

This calculation is performed while shifting a center point whichdevides the signal into two in order to compute a correlation function.By obtaining a peak of the correlation function, the position of thesymmetrical point of the image can be measured. In performing theposition measurement by these methods, as a method of performing theinterpolation, there is an interpolation method of using the positionmeasurement similar to that of a pre-process, such as filtering.

Specifically, the method consists of performing the measurement afterfiltering or interpolation is performed. For this case, the templateimage signal at the sampling intervals after the interpolation must beused in the normalized mutual correlation method and the like.

Moreover, there is also a method of using interpolation in anintermediate state of the measurement. Although both a differentialsignal and an original signal thereof are used in edge detection, sincethe differential signal is made for the purpose of “recognizing” an edgeposition, there are many cases where the interpolation is not required.

In this case, when a signal obtained by differentiating the input imagesignal and a signal subjected to interpolation and smoothing are used,there is a merit in the processing rate.

Furthermore, in the correlation method, the correlation function becomesa function of the relative position between the input image signals orbetween the input image signal and the template image signal, which aresignals typically sampled at the sampling intervals of the image data.

Accordingly, the interpolation is performed for this correlationfunction, and an improvement in the accuracy in the peak detection canalso be achieved.

Hereinbelow, a description of the simulation results of samplingmeasurement errors for the case of using the prior art and thisembodiment will be given.

FIG. 4 is a one-dimensional projection of a step difference mark imageformed on an image pickup device by an optical system for conditionswhere NA is 0.6, the wavelength is 0.6 μm, and the illumination sigma is1.0. This projection is obtained by a simulation.

When the minimum periodic component included in the image is calculatedin consideration of the magnification or the like of the optical systemand then converted with respect to wafer 12, the minimum periodiccomponent is 0.6/(2×0.6)=0.5 μm.

FIGS. 5 and 6 are signals obtained by sampling the image shown in FIG. 4in a sampling interval of 0.24 μm. The mark detection method accordingto one embodiment of the present invention was not used for obtainingthese results.

The difference between FIGS. 5 and 6 is the difference in the samplingphases in the simulation results. Specifically, in the actual apparatus,the simulation results correspond to measurement results for the case ofmoving the position of the image pickup device such as a CCD camera.

When FIGS. 5 and 6 are compared with each other, the shapes of thesignals are apparently different from each other, and thus it isanticipated that an error will occur when edge detection, thecorrelation measurement and the like are performed.

When the measurement position is changed due to the phase relationshipbetween the position of the image and the position of the image pickupdevice, the change in the measurement position causes an error in thesampling.

Therefore, in order to assess this error, after sampling is performedfor image data in a cycle of 0.24 μm while moving the same by a samplinginterval of 100 μm, edge detection is performed to investigate thedifference between the moving amount and the measurement value, and thenthe shape in FIG. 7 appears. FIG. 7 is a diagram showing the dislocationamount of the edge position for the case where the position of the imagefor the image pickup device is moved.

FIG. 15 shows a variation in the dislocation amount between the movingamount and the measurement value while changing the sampling interval.With reference to FIG. 15, although the variation amount behavescyclically, it can be seen that the error increases after the samplinginterval exceeds 0.2 μm. This is a condition where the sampling intervalis about 0.4 times the minimum periodic component of the image.

Incidentally, in the mark detection method according to one embodimentof the present invention, interpolation is performed during themeasurement.

FIGS. 8 and 9 are diagrams showing simulation results obtained byperforming a three-point interpolation on the respective sampled pointssampled from the signals shown in FIGS. 5 and 6.

In addition, FIG. 10 is a diagram showing the dislocation amount of theedge position for the case where the position of the image for the imagepickup device subjected to the interpolation is moved.

With reference to FIG. 10, there is hardly any dislocation amount of theedge position with respect to the amount of movement, and it can be seenrecognized that the accuracy is significantly improved by performing theinterpolation.

FIG. 15 is a diagram showing simulation results of the positionmeasurement error for the case where the sampling interval is changedwithout performing the pre-processing as described above.

In addition, FIG. 11 is a diagram showing simulation results of theposition measurement error for the case where the sampling interval ischanged when performing only interpolation.

FIG. 12 is a diagram showing simulation results of the positionmeasurement error for the case where the sampling interval is changedwhen performing only low-pass filtering.

FIG. 13 is a diagram showing simulation results of the positionmeasurement error for the case where the sampling interval is changedwhen performing interpolation and low-pass filtering.

With reference to FIG. 11, it can be seen that by performing thethree-point interpolation, the measurement accuracy is good until thesampling interval is 0.6 times the minimum periodic component of theimage, that is, until the sampling interval is 0.3 μm.

Under this condition, although aliasing occurs to diminish theoreticallythe measurement accuracy, the amplitude of the minimum periodiccomponent included in the image is not relatively large and contributeslittle to the measurement; thus the measurement accuracy is notdiminished very much. When the sampling interval exceeds 0.6 times theminimum periodic component of the image, the accuracy diminishes asshown in the drawing.

In FIGS. 12 and 13, filtering for removing a periodic component equal to1 μm or less is performed. Counting backwards fromP_(f)=1/(1/P_(s)−1/P_(min)), if P_(s)<0.33 μm, the aliasing componentwhich causes a problem will be removed ensuring good accuracy.

In the case of performing only low-pass filtering (see FIG. 12), theaccuracy is improved slightly in comparison with the case of performingno pre-processing (see FIG. 15); however, errors by sampling other thanaliasing are large, thus it cannot be said that there is a very goodeffect. It is recognized that the errors are shifted to a longersampling interval as a whole.

On the other hand, in the case of performing low-pass filtering andinterpolation together (see FIG. 11), the accuracy is improved incomparison with the case of only performing interpolation (see FIG. 12).With reference to FIG. 13, simulation results with extremely goodaccuracy can be obtained up until the sampling interval P_(s) reachesP_(s)=0.33 μm.

From the above simulations, it is recognized that the mark detectionmethod according to this embodiment is extremely effective for the casewhere the sampling interval cannot be set to be 0.5 times or less of theminimum periodic component of the image.

Herein, once more, a description will be made of the simulation resultsshown in FIG. 15 (numerical aperture NA=0.6, P_(min)=0.5 μm) and FIG. 16(numerical aperture NA=0.3, P_(min)=1 μm). As described above, theaccuracy is deminished for any NA in the region of0.2×P_(min)<P_(s)<0.39×P_(min) and in the region of P_(s)>0.41×P_(min).(The position measurement error becomes larger than the allowable errorof about 3 to 5 nm.)

However, the accuracy is improved in the region0.39×P_(min)<P_(s)<0.41×P_(min).

On the basis of the simulation results described above, if the pixelsize P_(s) of the image pickup device is determined so as to satisfy thecondition of 0.39×P_(min)<P_(s)<0.41×P_(min), the measurement error ofthe mark position (edge position) can be controlled within an allowablevalue (5 nm in FIG. 15), thus the measurement accuracy can be improved.In other words, for the case where the pixel size P_(s) cannot bechanged (in the case of a fixed pixel size), the magnification of theoptical system (image-forming optical system for observing a mark inorder to form an image thereof on the image pickup device) is determinedso as to satisfy the condition of 0.39×P_(min)<P_(s)<0.41×P_(min), thusthe measurement accuracy of the mark can be improved.

Moreover, on the basis of the foregoing simulation results, if thecondition that the line width of a mark=(2n+1)×P_(s)/2 is satisfiedbetween the shape (pitch) of an observed mark (mark having a cycle) andthe pixel size P_(s), that is, if the sampling position and the edgeposition are shifted from each other, the position measurement error canbe controlled to within the allowable error (the accuracy can beimproved). Accordingly, also by using a mark satisfying the conditionthat the line width of a mark=(2n+1)×P_(s)/2, the mark measurementaccuracy can be improved. In other words, when the magnification of theimage-forming optical system is determined so as to satisfy thecondition that the line width of a mark=(2n+1)×P_(s)/2, the markmeasurement accuracy can be improved. Note that this method is effectivefor both the condition P_(s)>0.2 P_(min), where interpolation isrequired, and the condition P_(s)>0.5 P_(min), where aliasing isgenerated.

Furthermore, by forming the line width of the mark unevenly in the mark,the sampling position and the edge position of the mark can be shiftedfrom each other. When such a mark is used for the measurement, the markmeasurement position can be improved. In addition, when such an unevenmark (a mark constituted of a plurality of lines and spaces and havingdifferent line widths and space widths in the visual field (within themark)) is used, the measurement error can be reduced in all the pixelsizes by an averaging effect. Note that this method is also effectivefor both the condition “P_(s)>0.2 P_(min), where interpolation isrequired, and the condition P_(s)>0.5 P_(min), where aliasing isgenerated.

Incidentally, the interpolation method in the above-described embodimentis a method of obtaining intensities (for example, luminance) ofpositions among the sample points (f₃ to f₆ in FIG. 2) by linearcombination of the intensities (for example, luminance) of the samplepoints.

More specifically, the interpolation method is a method of obtaining anarbitrary point (interpolation point f′₈) by performing a weightingoperation by the use of a proximate point (above-described f₅ in FIG. 2)of an arbitrary point to be subjected to interpolation (for example,previously described f′₈ in the description of FIG. 2) and severalsample points in front of and behind the proximate point(above-described f₄ and f₆ in FIG. 2).

The coefficients corresponding to the respective sample points (closesample points f₄ to f₆ in FIG. 2) used in this linear combination(weighting operation) are equivalent to the contribution from therespective sample points, that is, the weighting (weightingcoefficients). Note that negative values are also included in thisweighting.

Incidentally, according to the sampling theorem, in order to restorecontinuous data to its pre-sampled state, the condition in that the sumtotal of the weights for weighting is always constant must be satisfied.When an infinite number of sample points are used (when the sum of aninfinite number of coefficients is taken), the sum total of the weightscan be made constant, and thus the foregoing condition can be satisfied.

However, in the actual interpolation processing, the weighting operationis performed in a limited region (with limited sample points). When theinterpolation is performed in a limited region, the sum total of theweights is not constant and cyclical noise occurs at the samplingintervals, causing the reduction of an S/N ratio for a signal having alow contrast. In order to solve this problem, standardizing processingmust be performed on the above-described interpolation filter (weightingfunction, sinc function).

Hereinbelow, a description will be made for the standardizing processingfor the interpolation filter.

A description will be made for the case of standardizing theinterpolation filter S(dx) in the foregoing Equation (3), which is givenby the product of the interpolation filter S having a limited length inthe foregoing Equation (2) and the previously-described window functionW(dx).

First, before a value of a certain arbitrary point (the point to beinterpolated) is calculated by the use of the interpolation filter S(dx)in Equation (3), the sum (sum total) obtained by the interpolationfilter S(dx) is calculated.

Next, the coefficient values of the respective sample points(coefficients of the respective sample points obtained by theinterpolation filter) used for this operation are divided by theobtained sum total. Thus, in all the cases (in all the arbitraryinterpolation points), the sum total of the weighting can be set to “1”(a constant value). For this reason, the noise described above is notgenerated. Then, this result obtained by the division is used as aweight coefficient of the sample point.

Herein, a description will be specifically made for this standardizingprocessing by exemplifying the case where the interpolation is performedfor the arbitrary interpolation point f′₈ described with reference toFIG. 2 by the use of the sample points f₄, f₅ and f₆ in the vicinity ofinterpolation point f′₈. The weight coefficients of the respectivesample points f₄, f₅ and f₆, which are obtained by the interpolationfilter S(dx) are defined as a₄, a₅ and a₆. The sum total Σa of a₄ to a₆is obtained. And, by dividing the respective coefficients a₄ to a₆ bythe sum total Σa, a₄/Σa, a₅/Σa and a₆/Σa are determined as the weightcoefficients of the respective sample points f₄, f₅ and f₆.

Next, a description will be made for another aspect of the interpolationfilter.

Performing differentiation after performing interpolation or performinginterpolation after performing differentiation leads to an increase inthe processing time. Therefore, a description will be made for an aspectwhere differentiation and interpolation are simultaneously performedsimultaneously in order to achieve a shortening of the processing time.

Performing the differentiation calculus after performing filtering by asinc function S(dx) is equivalent to performing filtering by a functionf(x) represented by the following Equation (18) when the samplinginterval is defined as T. $\begin{matrix}{{f(x)} = {\frac{2f_{1}{\cos ( {2\pi \quad {x/2}T} )}}{x} - \frac{\sin ( {2\quad \pi \quad {x/2}T} )}{\pi \quad x^{2}}}} & (18)\end{matrix}$

where f(x)=0

In order to restore data by performing interpolation using this filterf(x), the condition that the sum total of the contributions (sum totalof the weight coefficients) for calculating a certain arbitraryinterpolation point is always zero (hereinafter, referred to as a“summation condition”) must be satisfied, and the condition that amoment Σf(x)·x is always constant (hereinafter, referred to as a “momentcondition”) must also be satisfied. However, as described previously,since the weighting operation is performed in a limited region (withlimited sample points), these two conditions cannot be satisfiednaturally. For this reason, when the interpolation is performed by theuse of this filter f(x) without satisfying the conditions, cyclicalnoise occurs at the sampling intervals, causing a reduction in an SINratio for a signal having a low contrast. Accordingly, in the case ofperforming filtering using this function f(x), the standardizingprocessing must also be performed.

First, the sum (sum total) Σf(x) of the coefficients in the respectivesample points obtained by the filter f(x) is calculated.

Next, the obtained sum total Σf(x) is divided by the number n of thesample points utilized for the linear combination (weighting operation),and Σf(x)/n obtained by the division is subtracted from the values ofthe coefficients of the respective sample points (coefficients of therespective sample points obtained by the filter f(x)). The function f(x)representing the respective coefficients is converted as described aboveto be a function f′(x), that is, f′(x)=f(x)−Σf(x)/n.

Next, by the use of the conversion function f′(x), the sum totalΣf′(x)·x of the moments f′(x)·x of the coefficients in the respectivesample points is obtained, and the respective coefficients by theconversion function f′(x) obtained previously are divided by this sumtotal of the moments. And, the result f′(x)/(Σf′(x)·x) obtained by thedivision is used as a weight coefficient of the sample point.

In the foregoing manner, standardization of the filter function f(x) ispossible, that is, the above-described summation condition and momentcondition can be satisfied.

Herein, a description will be specifically made for this standardizingprocessing by exemplifying the case where the interpolation is performedon the arbitrary interpolation point f′₈ described with reference toFIG. 2 using the sample points f₄, f₅ and f₆ in the vicinity of thepoint f′₈. Weight coefficients of the respective sample points f₄, f₅and f₆, which are obtained by the interpolation filter f(x), are definedas b₄, b₅ and b₆. The sum total Σb of these b₄ to b₆ is obtained. And,b₄−Σb/n, b₅−Σb/n and b₆−Σb/n, which are obtained by subtracting Σb/n,obtained by dividing the sum total Σb by the number of the sample points(in this case, three) from the respective coefficients b₄ to b₆, areobtained. These b₄−Σb/n, b₅−Σb/n, and b₆−Σb/n are respectively definedas b₄′, b₅′, and b₆′.

Next, the moments of the coefficients in the respective sample pointsare obtained. When the distances of the function f′(x) from the originin the sample points f₄, f₅ and f₆ are respectively defined as c₄, c₅and c₆, the moments of the respective coefficients b₄ to b₆ becomec₄·₄′, c₅·b₅′ and ″c₆·b₆′. Next, the sum total Σx·f′(x) of therespective moments is obtained (specifically, Σx·f′(x)=c₄·b₄′+c₅·b₅′+c₆·b₆′ is established).

Then, b₄′/Σx ·f′(x), b₅′/Σx·f′(x) and b₆′/Σx·f′(x), which are obtainedby dividing b₄′ to b₆′ obtained previously by the sum total Σx·f′(x) ofthe moments, are determined to be the weight coefficients of therespective sample points f₄, f₅ and f₆.

Note that although the alignment of the wafer has been exemplified inthe mark detection method and, the exposure method according to theabove-described embodiment of the present invention, the presentinvention is not limited to the above-described embodiment.

For example, in the position measurement using an image sensor having alimited minimum division unit such as a CCD or a line sensor, thepresent invention can be applied to position measurement for otherobjects such as a mask on a reference plate.

The method of interpolating for a discretized signal and the method ofusing a filter for removing aliasing is not limited to the case ofdetecting a position from an optical image.

Specifically, for example, like the above-described LSA-type alignmentsensor, in the case of performing position detection by taking a signalrelated to the alignment mark while moving (scanning) a wafer stagerelative to an alignment light, a light intensity signal for adiscretized position is similarly obtained. When detecting the positionfrom this signal, the sampling interval of the signal must besufficiently smaller than the minimum periodic component of the signal.The sampling interval is determined by the clock frequency of theelectrical system and the moving rate (scanning rate) of the waferstage. By using a large sampling interval to perform the interpolation,the scanning rate can be accelerated without reducing the accuracy.Therefore, not only can throughput be improved, but also the clockfrequency of the electrical system can be decreased to improve the S/Nratio.

In the case where the contrast of the image is measured at some pointswith respect to the direction of the optical axis of the projectionoptical system, and a focal point position measurement signal dependingon the focal point of the projection optical system is obtained, thebest focus position of the projection optical system is obtained, and afocal point position measurement signal regarding the discretized focalpoint is similarly obtained. Also, in this case, the number ofmeasurement points can be reduced by using the interpolation of thepresent invention for the focal point position measurement signalwithout lowering the measurement accuracy; thus the measurement rate canbe improved to a great extent.

Note that the exposure apparatus (FIG. 1) according to this embodimentis manufactured so as to be able to control the position of wafer 12with good accuracy at a fast rate, and to enable exposure with highexposure accuracy while improving throughput. Specifically, the exposureapparatus according to this embodiment is manufactured in such a mannerthat the respective elements shown in FIG. 1, such as theabove-described illumination systems 1 to 9, reticle alignment system31, wafer alignment system 32 and projection optical system 11 arecoupled and fabricated to one another electrically, mechanically, oroptically, and then an overall adjustment (electrical adjustment,operation confirmation, or the like) is performed thereto. Note that themanufacturing of the exposure apparatus is desirably performed in aclean room where the temperature, degree of cleanness, and the like arecontrolled.

Next, a description will be made for the manufacturing of a device usingthe exposure apparatus and method of this embodiment.

FIG. 21 is a flowchart for producing a device in this embodiment (asemiconductor chip such as an IC and an LSI, a liquid crystal panel, aCCD, a thin-film magnetic head, a micro-machine and the like). As shownin FIG. 21, first, in step S301 (designing step), the function of thedevice is designed (for example, design for a circuit of thesemiconductor device), and pattern design for realizing the function isperformed. Subsequently, in step S302 (mask manufacturing step), a maskhaving the designed circuit pattern formed thereon is manufactured.Meanwhile, in step S303 (wafer producing step), a wafer is produced bythe use of a material such as silicon.

Next, in step S304 (wafer processing step), an actual circuit or thelike is formed on the wafer by a lithography technique as describedlater using the mask and the wafer which are prepared in steps S301 toS303. Then, in step S305 (fabricating step), a chip is made using of thewafer processed in step S304. In this step S305, processes such as anassembly process (dicing, bonding) and a packaging process (chipsealing) are included.

Finally, in step S306 (inspection step), inspections, such as anoperation confirmation test and a durability test, are performed for thedevice fabricated in step S305. The device is completed after goingthrough such steps, and then shipped.

FIG. 22 is a flowchart showing the detailed flow of the foregoing stepS304 for the case of manufacturing a semiconductor device. In FIG. 22,the surface of the wafer is oxidized in step S311 (oxidation step). Instep S312 (CVD step), an insulating film is formed on the wafer surface.In step S313 (electrode forming step), an electrode is formed on thewafer by deposition. In step S314 (ion implanting step), ions areimplanted in the wafer. Each of the above steps S311 to S314 constitutesa pre-step for each stage of the wafer processing, and is selected andexecuted according to the necessary processing in each stage.

In each stage of the wafer processing, after the pre-steps arecompleted, a post-step is executed as follows. In this post-step, first,photosensitive agent is coated on the wafer in step S315 (resist formingstep). Then, in step S316 (exposure step), the circuit pattern of themask is printed and exposed on the wafer by the above-described exposureapparatus. Next, in step S317 (development step), the exposed wafer isdeveloped. Subsequently, in step S318 (etching step), exposed members ofa portion other than the portion where the resist remains are removed byetching. Then, in step S319 (resist removal step), the unnecessaryresist remaining after the etching has been completed is removed.

By performing these pre-steps and post-steps repeatedly, a plurality ofcircuit patterns are formed on the wafer.

In the above-described manner, a device having a fine pattern formedthereon is manufactured with high accuracy in mass production.

Note that the exposure apparatus of this embodiment can be applied to anexposure apparatus of a scanning type (U.S. Pat. No. 5,473,410), whichmoves a mask and a substrate synchronously to expose a pattern of amask. Moreover, the exposure apparatus of this embodiment can be appliedto an exposure apparatus of a step-and-repeat type, which exposes apattern of a mask in a state where the mask and substrate are madestatic, and then moves the substrate sequentially by steps. Furthermore,the exposure apparatus of this embodiment can be applied to a proximityexposure apparatus which allows a mask and a substrate to closelycontact each other without using a projection optical system, and thenexposes a pattern of the mask.

In addition, the application of the exposure apparatus is not limited toan exposure apparatus for manufacturing a semiconductor. For example,the exposure apparatus can be extensively applied to an exposureapparatus for liquid crystal which exposes a pattern of a liquid crystaldisplay device on a square-shaped glass plate, an exposure apparatus formanufacturing a thin-film magnetic head, and the like.

Not only a g-ray (436 nm), an i-ray (365 nm), a KrF excimer laser (248nm), an ArF excimer laser (193 nm) and an F₂ laser (157 nm), but also acharged particle ray such as an X-ray and an electron ray can be used asthe light source of the exposure apparatus of this embodiment. Forexample, in the case of using an electron ray, an electron gun, orlanthanum hexaboride (LaB₆) and tantalum (Ta) of the thermal electronradiation type can be used.

The magnification of the projection optical system is not only areduction system but may also be either an equal magnification system ora magnification system.

In the projection optical system, when using a far-ultraviolet ray suchas an excimer laser, a material which transmits a far-ultraviolet raytherethrough such as quartz and fluorite is used as the glass material.In the case of using an F₂ laser or an X-ray, the projection opticalsystem is a catadioptric or dioptric system (a reflection type reticleis also used). In addition, in the case of using an electron ray, anelectron optical system that consists of an electron lens and apolariscope may be satisfactorily used as the optic system. The opticalpath for transmitting an electron ray therethrough is in a vacuum state.

When using a linear motor for the wafer stage or the reticle stage (seeU.S. Pat. No. 5,623,853 or U.S. Pat. No. 5,528,1118), either anair-floating type utilizing an air-bearing or a magnetic-floating typeutilizing a Lorentz force or a reactance force may be used. Moreover,the stage may be either a type which moves along a guide or a guidelesstype which is not provided with a guide.

Regarding the apparatus for driving a stage, a plane motor for driving astage by an electromagnetic force may also be used, in which a magnetunit having magnets disposed in two dimensions, and an armature unithaving coils disposed in two dimensions, are opposite each other. Inthis case, it is satisfactory that any one of the magnetic unit and thearmature unit may be connected to the stage, and that the other unit maybe provided on a moving plane of the stage.

As disclosed in the gazette of Japanese Unexamined Patent Application,First Publication No. Hei 8-166475 (U.S. Pat. No. 5,528,118), a reactionforce generated by moving the wafer stage may be sent to the floor (theearth) mechanically by the use of a frame member.

As disclosed in Japanese Unexamined Patent Application, Firstpublication No. Hei 8-330224 (U.S. Ser. No. 08/416,558), a reactionforce generated by moving the reticle stage may be sent to the floor(the earth) mechanically by the use of a frame member.

In addition, in the above-described embodiment, a description has beenmade for the case where a sinc function is used when performing theinterpolation. However, the interpolation may be done by fitting using,for example, a spline function. In the case of performing theinterpolation on an input image signal itself, the interpolation may beperformed on a correlation function prepared on the basis of theconcerned image signal.

What is claimed is:
 1. A mark detection method comprising: irradiating amark formed on an object with a detection beam; forming an image of saidmark through an image-forming system; converting the image of said markformed on an image pickup element into an electrical image signal inorder to output a signal related to said electrical image signal atpredetermined sampling intervals; performing a smoothing operation toremove a component having a cycle being equal to or less than apredetermined cycle from said signal output at said sampling intervals;and performing interpolation on said signal related to said electricalimage signal in a cycle equal to or less than said predeterminedsampling interval, wherein said image pickup element has a predeterminedpixel size said predetermined sampling interval includes a cycle of saidpredetermined pixel size, said interpolation is performed in a cycleequal to or less than said predetermined pixel size, said pixel sizeP_(s) is a predetermined multiple of a minimum periodic componentP_(min) of an image formed on said image pickup element, and saidsmoothing operation removes a periodic component equal to or less than1/(1/P_(s)−1/P_(min)), on the basis of said predetermined pixel sizeP_(s) and said minimum periodic component P_(min).
 2. A mark detectionmethod according to claim 1, wherein said predetermined pixel size P_(s)is larger than 0.5 times of said minimum periodic component P_(min). 3.A mark detection method according to claim 2, wherein said smoothingoperation comprises: a step of setting a smoothing point where smoothingis performed for said image signal; a step of selecting, from said imagesignal, said image signal sampled in a predetermined range that includessaid smoothing point; a step of performing a sampling in a cycleidentical to the sampling interval of said image signal, with respect toa function which removes a periodic component smaller than said1/(1/P_(s)−1/P_(min)), according to a distance between a position ofsaid smoothing point and a position of said selected image signal, astep of calculating a product of said selected image signal and saidsampled function with respect to each of said image signals included insaid predetermined range, and adding them to each of said image signals.4. A mark detection method according to claim 2, wherein said smoothingoperation comprises: a step of setting an interpolation point whereinterpolation for said image signal is performed; a step of obtaining amost proximate position of said image signal, which is most proximate toa position of said interpolation point; a step of selecting, from saidimage signal, said image signal sampled in a predetermined rangeincluding said most proximate position; and a step of calculating aproduct of said selected image signal and a function for removing aperiodic component smaller than said 1/(1/P_(s)−1/P_(min)) according toa distance to a position of said selected image signal with respect toeach of said image signals included in said predetermined range, andadding said product to said each of said image signals.
 5. A markdetection method according to claim 1, wherein said image signal isoutput as a sample point in said predetermined sampling interval, andinterpolation is performed for an arbitrary point in a cycle equal to orless than said predetermined sampling interval by an interpolationmethod using a conversion including a linear combination of a pluralityof said sample points located in the vicinity of the arbitrary point. 6.A mark detection method according to claim 5, wherein said interpolationmethod includes a weighting operation by the use of said plurality ofsaid sample points.
 7. A mark detection method according to claim 5,wherein measurement is performed for a position of said object on thebasis of said interpolated image signal.
 8. A mark detection methodaccording to claim 5, wherein said predetermined sampling interval isdetermined on the basis of an amount of position measurement erroramount in said measurement.
 9. A mark detection method according toclaim 8, wherein said object is a substrate onto which a circuit patternis transferred, and said amount of position measurement error in saidpredetermined sampling interval has a value of one-fourth a minimum linewidth of said circuit pattern on which the total overlay is transferredonto said substrate.
 10. A mark detection method according to claim 1,wherein said interpolation is performed on said image signal itself. 11.A mark detection method comprising: imaging a mark formed on an object;converting an image of said mark, which is formed on an image pickupelement, into an electrical image signal; outputting a signal related tosaid image signal as a sample point at predetermined sampling intervals;performing interpolation on an arbitrary point in a cycle equal to orless than said predetermined sampling interval by an interpolationmethod using a conversion including a linear combination of a pluralityof said sample points; and standardizing an interpolation filter whichdetermines each weighting coefficient used in said weighting operation,so that the total of the weighting coefficients used when interpolationis performed on a first arbitrary point and the sum total of theweighting coefficients used when interpolation is performed on a secondarbitrary point different from the first arbitrary point can bepredetermined values.
 12. A mark detection method according to claim 11,wherein said interpolation method includes performing a weightingoperation by the use of said plurality of said sample points located inthe vicinity of said arbitrary point.
 13. A mark detection methodaccording to claim 11, wherein when said predetermined sampling intervalis T, and said interpolation filter includes an interpolation functions(dx) represented as:${s( {x} )} = {\frac{\sin ( {2\pi \quad {{x}/2}T} )}{2\pi \quad {{x}/2}T}.}$


14. A mark detection method according to claim 13, wherein saidinterpolation filter S(dx) is represented as S(dx)=s(dx)·W(dx), which isa product of said interpolation function s(dx) and a window functionW(dx) converging an end portion of said interpolation function s(dx) tozero.
 15. A mark detection method according to claim 14, wherein when alength of a window is R, said window function W(dx) is represented as:${W( {x} )} = \frac{1 + {\cos ( {2\pi \quad {{x}/R}} )}}{2}$


16. A mark detection method according to claim 11, wherein saidstandardization converts the respective coefficients of saidinterpolation filter by dividing the respective coefficients by the sumtotal of the respective coefficients.
 17. A mark detection methodaccording to claim 11, wherein a smoothing processing of removing aperiodic component equal to or less than a predetermined cycle from saidimage signal output as a sample point at said predetermined samplingintervals.
 18. A mark detection method according to claim 17, whereinsaid image pickup element has a pixel size P_(s) a predeterminedmultiple of a minimum periodic component P_(min) of an image formed onsaid image pickup element, and said smoothing processing includesremoving a periodic component equal to or less than1/(1/P_(s)−1/P_(min)), on the basis of said pixel size P_(s) and saidminimum periodic component P_(min).
 19. A mark detection methodaccording to claim 18, wherein said pixel size P_(s) is larger than 0.5times of said minimum periodic component P_(min).
 20. A mark detectionmethod according to claim 11, wherein said interpolation is performed onsaid image signal itself.
 21. A mark detection method according to claim11, wherein said image pickup element has a predetermined pixel size,said predetermined sampling interval includes a cycle of saidpredetermined pixel size, and said interpolation is performed in a cycleequal to or less than said predetermined pixel size.
 22. A markdetection method according to claim 21, wherein said pixel size is apredetermined times of a minimum periodic component of an image formedon said image pickup element.
 23. A mark detection method according toclaim 22, wherein said mark is irradiated with a detection beam, and animage of said mark is formed through an image-forming system, saidminimum periodic component is defined by λ/2NA on the basis of awavelength λ of said detection beam and the numerical aperture NA ofsaid image-forming system.
 24. A mark detection method according toclaim 23, wherein said pixel size is equal to or more than 0.2 times ofsaid minimum periodic component.
 25. A mark detection method accordingto claim 24, wherein said pixel size is equal to or less than 0.5 timesof said minimum periodic component.
 26. A mark detection methodaccording to claim 25, wherein said pixel size is equal to or more than0.39 times of said minimum periodic component, or is equal to or lessthan 0.41 times of said minimum periodic component.
 27. A mark detectionmethod according to claim 11, wherein the position of said object ismeasured based on said interpolated image signal.
 28. A mark detectionmethod according to claim 27, wherein said predetermined samplinginterval is determined based on an amount of position measurement errorof said measurement.
 29. A mark detection method according to claim 27,wherein said object is a substrate onto which a circuit pattern istransferred, and said amount of position measurement error in saidpredetermined sampling interval has a value of one-fourth a minimum linewidth of said circuit pattern on which the total overlay is transferredonto said substrate.
 30. A mark detection method comprising: irradiatinga mark formed on an object with a detection beam; forming an image ofsaid mark through an image-forming system; converting the image of saidmark formed on an image pickup element into an electrical image signalin order to output a signal related to said electrical image signal atpredetermined sampling intervals; and performing interpolation on saidsignal related to said electrical image signal in a cycle equal to orless than said predetermined sampling interval; wherein saidinterpolation is performed with respect to a correlation functionobtained based on said image signal.
 31. An exposure method, whereinsaid object is a substrate onto which a predetermined pattern istransferred, and said predetermined pattern is transferred onto saidsubstrate aligned on the basis of a mark detected by the use of saidmark detection method according to claim
 1. 32. An exposure method,wherein said object is a substrate onto which a predetermined pattern istransferred, and said predetermined pattern is transferred onto saidsubstrate aligned on the basis of a mark detected by the use of saidmark detection method according to claim
 11. 33. An exposure method,wherein said object is a substrate onto which a predetermined pattern istransferred, and said predetermined pattern is transferred onto saidsubstrate aligned on the basis of a mark detected by the use of saidmark detection method according to claim
 30. 34. A device manufacturingmethod, wherein a device is manufactured by the use of said exposuremethod of transferring said predetermined pattern onto said substrateaccording to claim
 31. 35. A device manufacturing method, wherein adevice is manufactured by the use of said exposure method oftransferring said predetermined pattern onto said substrate according toclaim
 32. 36. A device manufacturing method, wherein a device ismanufactured by the use of said exposure method of transferring saidpredetermined pattern onto said substrate according to claim
 33. 37. Amark detection apparatus comprising: an irradiation system whichirradiates a mark formed on an object with a detection beam; animage-forming system which forms an image of a mark on an image-formingsurface; an image pickup element disposed on said image-forming surface;a sampling device, which is electrically connected to the image pickupelement, and which converts the image of said mark into an electricalimage signal in order to output a signal related to the image signal ina predetermined sampling interval; a smoothing device, which iselectrically connected to the sampling device, and which removes aperiodic component equal to or less than a predetermined cycle from asignal output at said sampling intervals by said sampling device; and aninterpolation device, which is electrically connected to the smoothingdevice, and which interpolates the smoothed sign related to said imagesignal in a cycle equal to or less than said predetermined samplinginterval, wherein said image pickup element has a pixel size P_(s) whichis a predetermined multiple of a minimum periodic component P_(min) ofan image formed on said image pickup element, and said smoothing deviceremoves a periodic component equal to or less than1/(1/P_(s)−1/P_(min)), on the basis of said pixel size P_(s) and saidminimum periodic component P_(min).
 38. A mark detection apparatusaccording to claim 37, wherein said pixel size P_(s) is larger than 0.5times of said minimum periodic component P_(min).
 39. A mark detectionapparatus according to claim 37, wherein said sampling device outputssaid image signal at said predetermined sampling intervals, and saidinterpolation device performs interpolation with respect to said imagesignal.
 40. An exposure apparatus comprising a mark detection apparatusaccording to claim 37, wherein said object is a substrate to betransferred with a predetermined pattern, and said exposure apparatustransfers said predetermined pattern onto said substrate positionedbased on a mark detected by said mark detection apparatus.
 41. A devicemanufactured through a step of transferring said predetermined patternonto said substrate by said exposure apparatus according to claim 40.42. A mark detection apparatus comprising: a sampling device which picksup an image of a mark formed on an object, converts said image of saidmark into an electrical image signal, and outputs a signal related tosaid image signal at predetermined sampling intervals; an interpolationdevice, which is electrically connected to the sampling device, andwhich interpolates an arbitrary point having a cycle equal to or lessthan said predetermined sampling interval by an interpolation methodincluding a weighting operation using said plurality of said samplepoints located in the vicinity of said arbitrary point; and astandardizing device, which is electrically connected to theinterpolation device, and which standardizes an interpolation filterwhich determines each weighting coefficient used in said weightingoperation, so that the total of the weighting coefficients used wheninterpolation is performed on a first arbitrary point and the sum totalof the weighting coefficients used when interpolation is performed on asecond arbitrary point different from the first arbitrary point can bepredetermined values.
 43. A mark detection apparatus according to claim42, wherein when said predetermined sampling interval is T, and saidinterpolation filter includes an interpolation function s(dx)represented as:${s( {x} )} = \frac{\sin ( {2\pi \quad {{x}/2}T} )}{2\pi \quad {{x}/2}T}$


44. A mark detection apparatus according to claim 42, wherein said markdetection apparatus irradiates said mark with a detection beam, andforms an image of said mark through an image-forming system; said imagepickup element has a said pixel size which is a predetermined multipleof a minimum periodic component of an image formed on said image pickupelement; said minimum periodic component is defined by λ/2NA on thebasis of a wavelength λ of said detection beam and the numericalaperture NA of said image-forming system; and said pixel size is equalto 0.2 times or more and 0.5 times or less of said minimum periodiccomponent.
 45. A mark detection apparatus according to claim 42, whereinsaid sampling device outputs said image signal at said predeterminedsampling intervals, and said interpolation device performs interpolationwith respect to said image signal.
 46. An exposure apparatus comprisinga mark detection apparatus according to claim 42, wherein said object isa substrate to be transferred with a predetermined pattern, and saidexposure apparatus transfers said predetermined pattern onto saidsubstrate positioned based on a mark detected by said mark detectionapparatus.
 47. A device manufactured through a step of transferring saidpredetermined pattern onto said substrate by said exposure apparatusaccording to claim
 46. 48. A mark detection device comprising: anirradiation system which irradiates a mark formed on an object with adetection beam; an image-forming system which forms an image of a markon an image-forming surface; a sampling device which includes an imagepickup element disposed on said image-forming surface, converts saidimage of said mark into an electrical image signal, and outputs a signalrelated to said image signal at a predetermined sampling intervals; andan interpolation device, which is electrically connected to the samplingdevice, and which performs interpolation of said signal related to saidelectrical image signal in a cycle equal to or less than saidpredetermined sampling interval; wherein said interpolation deviceperforms said interpolation with respect to a correlation functionobtained based on said image signal.
 49. A mark detection apparatusaccording to claim 48, wherein said sampling device outputs said imagesignal at said predetermined sampling intervals, and said interpolationdevice performs interpolation with respect to said image signal.
 50. Anexposure apparatus comprising a mark detection apparatus according toclaim 48, wherein said object is a substrate to be transferred with apredetermined pattern, and said exposure apparatus transfers saidpredetermined pattern onto said substrate positioned based on a markdetected by said mark detection apparatus.
 51. A device manufacturedthrough a step of transferring said predetermined pattern onto saidsubstrate by said exposure apparatus according to claim 50.